Derivative-free algorithm for computing zeros of analytic functions

Computing (Vienna/New York)

Volumn 63, Issue 1, 1999, Pages 69-91

  Kravanja, P ^a   Van Barel, M ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; POLYNOMIALS;

FORMAL ORTHOGONAL POLYNOMIALS; QUADRATURE METHOD;

POLES AND ZEROS;

EID: 0032680260     PISSN: 0010485X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s006070050051     Document Type: Article

References (32)

1. Bojanczyk, A. W., Heinig, G.: A multi-step algorithm for Hankel matrices. J. Complexity 10, 142-164 (1994).
2. Botten, L. C., Craig, M. S., McPhedran, R. C.: Complex zeros of analytic functions. Comput. Phys. Commun. 29, 245-259 (1983).
3. Boult, T., Sikorski, K.: An optimal complexity algorithm for computing the topological degree in two dimensions. SIAM J. Sci. Stat. Comput. 10, 686-698 (1989).
4. Bultheel, A., Van Barel, M.: Linear algebra, rational approximation and orthogonal polynomials. Amsterdam: North-Holland 1997.
5. Cabay, S., Meleshko, R.: A weakly stable algorithm for Padé approximants and the inversion of Hankel matrices. SIAM J. Matrix Anal. Appl. 14, 735-765 (1993).
6. Carpentier, M. C., Dos Santos, A. F.: Solution of equations involving analytic functions. J. Comput. Phys. 45, 210-220 (1982).
7. Davies, B.: Locating the zeros of an analytic function. J. Comput. Phys. 66, 36-49 (1986).
8. Delves, L. M., Lyness, J. N.: A numerical method for locating the zeros of an analytic function. Math. Comput. 21, 543-560 (1967).
9. Draux, A.: Polynômes orthogonaux formels - applications. Berlin Heidelberg New York: Springer 1983.
10. Draux, A.: Formal orthogonal polynomials revisited. Applications. Numer. Alg. 11, 143-158 (1996).
11. Freund, R. W., Zha, H.: A look-ahead algorithm for the solution of general Hankel systems. Numer. Math. 64, 295-321 (1993).
12. Gautschi, W.: On the construction of Gaussian quadrature rules from modified moments. Math. Comput. 24, 245-260 (1970).
13. Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3, 289-317 (1982).
14. Gautschi, W.: On the sensitivity of orthogonal polynomials to perturbations in the moments. Numer. Math. 45, 369-382 (1986).
15. Gautschi, W.: How (un)stable are Vandermonde systems? In: Asymptotic and computational analysis: conference in honor of Frank W. J. Olver's 65th Birthday (Wong, R., ed.), pp. 193-210. New York: Marcel Dekker 1990.
16. Gautschi, W., Inglese, G.: Lower bounds for the condition number of Vandermonde matrices. Numer. Math. 52, 241-250 (1988).
17. Gragg, W. B., Gutknecht, M. H.: Stable look-ahead versions of the Euclidean and Chebyshev algorithms. In: Approximation and computation: A festschrift in honor of walter Gautschi (Zahar, R. V. M., ed.), pp. 231-260. Basel: Birkhäuser 1994.
18. Gutknecht, M. H.: A completed theory of the unsymmetric Lanczos process and related algorithms, Part I. SIAM J. Matrix Anal. Appl. 13, 594-639 (1992).
19. Gutknecht, M. H.: A completed theory of the unsymmetric Lanczos process and related algorithms, Part II. SIAM J. Matrix Anal. Appl. 15, 15-58 (1994).
20. Henrici, P.: Applied and computational complex analysis: I. Power series - integration - conformai mapping - location of zeros. New York: Wiley 1974.
21. Herlocker, J., Ely, J.: An automatic and guaranteed determination of the number of roots of an analytic function interior to a simple closed curve in the complex plane. Reliable Comput. 1, 239-250 (1995).
22. Ioakimidis, N. I.: Quadrature methods for the determination of zeros of transcendental functions - a review. In: Numerical integration: recent developments, software and applications (Keast, P., Fairweather, G., eds.), pp. 61-82. Dordrecht: Reidel 1987.
23. Ioakimidis, N. I., Anastasselou, E. G.: A modification of the Delves-Lyness method for locating the zeros of analytic functions. J. Comput. Phys. 59, 490-492 1985.
24. Kravanja, P., Sakurai, T., Van Barel, M.: On locating clusters of zeros of analytic functions. BIT 39, (1999) (to appear).
25. Kravanja, P., Van Barel, M., Ragos, O., Vrahatis, M. N., Zafiropoulos, F. A.: ZEAL: A mathematical software package for computing zeros of analytic functions. Comput. Phys. Commun. (to appear).
26. Li, T.-Y.: On locating all zeros of an analytic function within a bounded domain by a revised Delves/Lyness method. SIAM J. Numer. Anal. 20, 865-871 (1983).
27. Lloyd, N. G.: Degree theory. Cambridge: Cambridge University Press 1978.
28. Traub, J. F.: Wasilkowski, G. W., Woźniakowski, H.: Information-Based Complexity. New York: Academic Press, 1988.
29. Watson, G. N.: A treatise on the theory of Bessel functions. Cambridge: Cambridge University Press 1966.
30. Wilkinson, J. H.: The evaluation of the zeros of ill-conditioned polynomials. Part I. Numer. Math. 1, 150-166 (1959).
31. Ying, X.: A reliable root solver for automatic computation with application to stress analysis of a composite plane wedge. D. Sc. dissertation, Washington University in St. Louis, 1986.
32. Ying, X., Katz, I. N.: A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain. Numer. Math. 53, 143-163 (1988).